Connection manifold

A connection manifold always consists of a topological manifold together with a connection $Γ$.

However, often there is an implicitly assumed (default) connection, like the LeviCivitaConnection connection on a Riemannian manifold. It is not necessary to use this decorator if you implement just one (or the first) connection. If you later introduce a second, the old (first) connection can be used without an explicitly stated connection.

This manifold decorator serves two purposes:

1. to implement different connections (e.g. in closed form) for one AbstractManifold
2. to provide a way to compute geodesics on manifolds, where this AbstractAffineConnection does not yield a closed formula.

• Connection manifold
• • Functions
  • Charts and bases of vector spaces

Functions

Charts and bases of vector spaces

All connection-related functions take a basis of a vector space as one of the arguments. This is needed because generally there is no way to define these functions without referencing a basis. In some cases there is no need to be explicit about this basis, and then for example a DefaultOrthonormalBasis object can be used. In cases where being explicit about these bases is needed, for example when using multiple charts, a basis can be specified, for example using induced_basis.